Test for average of independent, non-Identical binomial distributions

I have a box with $$m$$ coins, each with a probability $$p_i$$ for $$1\leq i \leq m$$ of flipping heads. As an experiment, I flip each coin $$n_i$$ times, recording the results.

What is a statistical test I can perform against the hypothesis that the coins are fair on average? (i.e., that $$(\sum p_i)/m = 0.5$$, or that if you choose a coin uniformly at random and flip it the probability of heads is 0.5)

For a single coin, you can use the binomial test, but I'm not sure how to adapt it for this situation -- it seems like you should be able to do so using some weighting scheme. Or, is there an approximation that would be appropriate? In my application, $$m$$ is fairly small, and the $$n_i$$ and $$p_i$$ are likely quite similar among themselves.

• For homework problems, please add the self-study tag. – StatsStudent Jan 9 at 14:57
• In fact, this is related to a public health research question. I restated it with coins to make it concise. I do think the answer is probably not complicated but I don't immediately know what it is. – rwgp Jan 9 at 19:26
• If you are interested in the global null hypothesis of p=0.5 then a binomial test is OK, you would just aggregate all the data and check whether the global proportion is equal to 0.5. Alternatively a $\chi^2$ test. – user2974951 Jan 10 at 11:35
• Yeah, I realize that about the global hypothesis, and I explicitly don't care about that -- e.g., the coins could have been flipped differing numbers of times, and therefore a global p=0.5 could be accounted for be differences in number of flips. If I knew that all the flip counts were the same, this would be equivalent to my question. Is there some way to resample so that all would have the same number of flips? – rwgp Jan 18 at 21:44